Bipartite Turán Number for Trees

• The paper introduces the bipartite Turán number for trees, defining maximum edge counts in F-free subgraphs of complete bipartite graphs with precise formulas.
• Explicit results for small trees, double stars, spiders, and paths showcase innovative extremal constructions and combinatorial proof techniques.
• The study extends classical Turán theory to connected bipartite graphs, establishing asymptotic ratios and outlining open problems in forbidden subgraph theory.

A bipartite Turán number for trees encapsulates the extremal function governing the maximal edge count in bipartite host graphs that avoid specified tree substructures. For a fixed bipartite forbidden graph $F$, and integers $1 \leq a \leq b$, the bipartite Turán number $\exb(a,b,F)$ is defined as the maximum number of edges in a $F$-free subgraph of the complete bipartite graph $K_{a,b}$; its connected version, $\exb,c(a,b,F)$, restricts to connected subgraphs. Systematic investigations have yielded sharp bounds, explicit formulas, and precise extremal configurations for diverse classes of trees, particularly emphasizing small trees, double stars, spiders, paths, and related structures (Caro et al., 13 Feb 2025, Bonamy et al., 10 Nov 2025, He et al., 11 Nov 2025).

1. Formal Definitions and Variants

Let $F$ be a fixed bipartite graph. The bipartite Turán number is defined by: $\exb(a,b,F) = \max\{e(G) : G \subseteq K_{a,b},\; F \not\subseteq G\},$ where $K_{a,b}$ denotes the complete bipartite graph with parts of size $a$ and $b$. The balanced case is denoted $\exb(n,F) = \exb(n,n,F)$.

The variant requiring connectivity is: $\exb,c(a,b,F) = \max\{e(G) : G \subseteq K_{a,b},\; G\text{ connected},\; F \not\subseteq G\},$ with the corresponding abbreviation for balanced graphs.

For a family $\mathcal F$ of forbidden graphs, definitions extend verbatim to $\exb(a,b,\mathcal F)$ and $\exb,c(a,b,\mathcal F)$. For paths $P_k$ and arbitrary trees $T$, refinements in the connected setting are essential for sharp extremal results (Caro et al., 13 Feb 2025, Bonamy et al., 10 Nov 2025, He et al., 11 Nov 2025).

2. Classical Bounds and Comparisons

2.1 Comparison to Classical Turán Numbers

The classical Turán number $\ex(N,F)$, for $F$-free graphs on $N$ vertices, satisfies: $\ex(n,F) < \exb(n,F) \leq \ex(2n,F)$ for all $n$ and $F$ (Caro et al., 13 Feb 2025). Construction methods establish

$\exb(n,F) \geq (2+o(1))\,\ex(2n,F)$

with $o(1) \to 0$ as $n \to \infty$.

2.2 Degree and Partition-Based Lower Bounds

For $F$ of maximum degree $\Delta = \Delta(F)$: $\begin{cases} \Delta \ge 2 \implies \exb(n,F) \ge (\Delta-1)n,\ \Delta \ge 3 \implies \exb,c(n,F) \ge (\Delta-1)n,\ \Delta = 1, F\neq K_2 \implies \exb(n,F) \ge n. \end{cases}$ For bipartite $F$ with parts $s \leq L$: $\begin{aligned} \exb(a,b,F) &\geq (s-1)\bigl((a-s+1)+(b-s+1)\bigr),\ \exb(n,F) &\geq 2(s-1)(n-s+1),\ \exb(a,b,F) &\geq (L-1)\min\{a,b\},\quad \exb(n,F) \geq (L-1)n. \end{aligned}$

3. Exact Values for Small Trees, Double Stars, Spiders, and Paths

3.1 Trees with Up to Six Vertices

Explicit formulas have been established for all trees of order $\le 6$ (Caro et al., 13 Feb 2025):

Tree $T$	$\exb(n,T)$ Formula
$P_4$	$2n-2$ ($n\ge2$)
$P_5$	$2n-(n\bmod 2)$
$P_6$	$6$ for $n=3$, $4n-8$ for $n\ge4$
$K_{1,3}$	$2n$
$K_{1,4}$	$3n$
$S_{2,1,1}$	$2n$

Six‐vertex spiders and double stars are also resolved:

• $S_{3,1,1}$: $\exb(n,S_{3,1,1}) \le 3n$, with equality iff $3\mid n$.
• $S_{2,2,1}$: $\exb(n,S_{2,2,1}) = 4n-8$ for $n\ge4$, $\exb(3,S_{2,2,1}) = 6$.
• $D_{2,2}$: Piecewise formula with extremal graphs as unions of complete bipartite blocks.

3.2 Double Stars and Spiders

For a double star $D_{s,t}$, the critical value $f(s,t)=4(s+1)^3$ partitions cases:

• If $a,b > f(s,t)$, $s< t<2s$:

$\exb(a,b,D_{s,t})=s(a+b-2s)$

with the extremal graph being $K_{s,a-s}\cup K_{s,b-s}$.

• If $2s\le t$:

$\exb(a,b,D_{s,t}) \le \frac{t}{2}(a+b)$

attained for $a=b$ divisible by $t$.

For spiders $S_{a_1,\dots,a_j}$:

• The “star+leaf” spider $S_{2,d+1}$ yields

$\exb(a,b,S_{2,d+1}) = \max\{da, d(a-1)+ (b-a+1)\}$

• For $S_{3,d+1}$ with large $n$:

$\exb(n,S_{3,d+1}) = (d+1) n - c_{d,n}$

$c_{d,n}$ accounts for divisibility.

3.3 Paths and General Trees

For any $k\ge3$, and $a,b \ge k$, for both $P_{2k-1}$ and $P_{2k}$ (Bonamy et al., 10 Nov 2025, He et al., 11 Nov 2025): $\ex_{b,c}(a,b,P_{2k-1}) = \ex_{b,c}(a,b,P_{2k}) = (k-2)(b-1) + a$ In balanced case ($a=b=n$): $\ex_{b,c}(n,n,P_\ell) = (k-2)(n-1) + n = (k-1)n - (k-2)$ for $\ell\in\{2k-1,2k\}$; for $\ell\in\{5,6\}$, one recovers $2n-1$.

Extremal graphs for these cases consist of a $K_{k-2,b}$ component plus leaves.

4. Extremal Graph Constructions and Proof Techniques

Extremal constructions rely on combinatorial assembly of bipartite blocks and leaf attachments:

• For paths and certain spiders: complete subgraphs form the “core,” while leaves attach to minimize the risk of forbidden substructures.
• Double star extremals: decomposition into suitable complete bipartite subgraphs.
• For long cycles: methods draw from Kopylov’s technique using vertex deletion and Jackson’s lemma adapted for the bipartite case.

Inductive proofs use deletion of low-degree vertices and block-tree arguments. Weight-counting and fractional edge assignments are deployed for upper bounds in double star and spider contexts (Caro et al., 13 Feb 2025, Bonamy et al., 10 Nov 2025, He et al., 11 Nov 2025).

5. Solving the Yuan–Zhang Problem for Bipartitions

For the family $\mathcal T_{k,\ell}$ of all trees whose bipartition has parts of sizes $k$ and $\ell$ (Caro et al., 13 Feb 2025):

• If $k\leq\ell\leq2k-1$ and $a>4k^3$:

$\exb(a,b,\mathcal T_{k,\ell}) = (k-1)(a+b-2(k-1))$

with the unique extremal graph $K_{k-1,a-k+1}\cup K_{k-1,b-k+1}$.

• If $\ell\geq 2k$:

$\exb(a,b,\mathcal T_{k,\ell}) < (\ell-1)(a+b)$

and for $a=b=n$ divisible by $\ell-1$ the bound is sharp.

6. Asymptotic Behavior and Ratios

Define $|T|$ as the order of the tree $T$. Asymptotic ratios for worst-case extremal counts are: $Y_{b,T} = \limsup_{n\to\infty} \frac{\exb(n,T)}{(|T|-2)\,n}$

$y_b = \inf_{T:|T|\ge k} Y_{b,T},\quad k\to\infty$

$Y_{b,c,T} = \limsup_{n\to\infty} \frac{\exb,c(n,T)}{(|T|-2)\,n},\quad y_{b,c} = \inf_{T:|T|\ge k} Y_{b,c,T},\quad k\to\infty$

It has been shown: $y_b = \frac{2}{3}$

$y_{b,c} \ge 0.207,\quad \inf_T \frac{\exb,c(n,T)}{\exb(n,T)} \ge 0.1035$

These bounds are established via explicit extremal constructions and quadratic optimization (Caro et al., 13 Feb 2025).

7. Connected Bipartite Turán Problem for Long Cycles and Paths

The exact structure and edge count for $P_k$-free connected bipartite graphs with color class sizes $m\le n$ is:

• If $m<\tfrac{k}{2}$: $|E(G)| \le mn$
• If $m\ge\tfrac{k}{2}\ge2$:

$|E(G)| \le t_k n + (m-t_k)$

where $t_k = \lfloor (k-3)/2 \rfloor$, with equality attained by a unique construction $B_1(m,n,k)$ (core $K_{t_k, n}$ plus leaves) (He et al., 11 Nov 2025).

These results generalize classical Turán-type problems for bipartite graphs, providing deep connections to cycle-free and path-free graph theory via classical theorems of Gyárfás–Rousseau–Schelp and Jackson.

8. Generalizations, Applications, and Open Problems

Extensions to brooms and related structures have established sharp edge bounds for the corresponding bipartite Turán numbers (Bonamy et al., 10 Nov 2025). Nevertheless, the general determination of $\exb,c(a,b,T)$ for arbitrary trees $T$ remains partially unresolved. The structural features of $T$ that dictate the linear extremal function in terms of $(a,b)$ are an open problem of active interest.

Key techniques such as local switching, maximum-cut arguments, inductive vertex deletion, regular subgraph embeddings, and cycle-and-chord-avoidance demonstrate the richness of the field and its connections to advanced extremal graph theory. These investigations lay the foundation for further combinatorial advances and applications in forbidden subgraph theory across bipartite classes.